The present study focuses both on the influence of impact scale on ejecta expansion and on specific features of ejecta deposits around relatively small craters (i.e., those a few kilometers in width). The numerical model is based on the SOVA multimaterial multidimensional hydrocode, considering subaerial vertical impacts only, applying a 2-D version of the code to projectiles of 100, 300, and 1000 m diameter. Ejecta can roughly be divided into two categories: “ballistic” ejecta and “convective” ejecta; the ballistic ejecta are the ejecta with which the air interacts only slightly, while the convective ejecta motion is entirely defined by the air flow. The degree of particle/air interaction can be defined by the time/length of particle travel before deceleration. Ejecta size-distributions for the impacts modeled can be described by the same power law, but the size of maximum fragment increases with scale. There is no qualitative difference between the 100 m diameter projectile case and the 300 m diameter projectile impact. In both cases, fine ejecta decelerate in the air at a small distance from launching point and then rise to the stratosphere by air flows induced by the impacts. In the 1000 m-scale impact, the mass of ejecta is so large that it moves the atmosphere itself to high altitudes. Thus, the atmosphere cannot decelerate even the fine ejecta and they consequently expand to the rarefied upper atmosphere. In the upper atmosphere, even fine ejecta move more or less ballistically and therefore may travel to high altitudes.

Impact studies, especially those concerning ejecta distributions, have most commonly been carried out on large craters (Stöffler et al. 2002; Dypvik et al. 2006; Artemieva and Morgan 2009), but lately ejecta distribution information on smaller impact craters has also been presented; e.g., Suuroja and Suuroja (2006) on the 4 km diameter Kärdla structure, Maloof et al. (2010) on the 1.88 km diameter Lonar crater, and Housen et al. () on the 2.7 km diameter Ritland structure (Table 1). In the present paper, we focus on the influence of impact scale (i.e., projectile size and/or energy), on ejecta expansion, and on specific features of ejecta deposits around relatively small craters (i.e., those a few kilometers in diameter). The impact scale influences both the processes of ejection and those of ejecta expansion, which are strongly controlled by atmospheric drag. In small events, the total mass of ejecta is minor compared with the atmospheric mass disturbed by the impact plume and ejecta decelerate over short distances, whereas in large events, the ejecta may penetrate to high altitudes/distances (Artemieva and Morgan 2009).

Table 1. Information on selected terrestrial impact craters

Crater

Crater diameter (km)

Estimated bolide size (m)

Lonar

1.9

60

Ritland

2.7

115

Gardnos

5.0

250

Kärdla

7.0

400

Lockne

7.5

400

Ries

26

1200

Mjølnir

40

1600

Chicxulub

150

10000

To study these issues, we have performed numerical modeling of three theoretical cases of vertical impact with 100, 300, and 1000 m diameter bolides, respectively. The aim has been to study both formation and distribution of impact ejecta, with special emphasis on the effect of atmospheric drag.

Ejecta expansion has only been theoretically considered in relatively few cases, by taking into account its interaction with air (atmospheric drag). These studies refer to specific (mainly large) craters, like 30 km in diameter Ries (Stöffler et al. 2002), more than 100 km in diameter Chicxulub (Artemieva and Morgan 2009), and 7 km in diameter Lockne (Lindström et al. 2005). Some theoretical aspects of the problem were considered analytically in Schultz and Gault (1979). In this paper, most attention has been concentrated on modeling of ejecta expansion in smaller impacts. The aim of the present study was to consider how the atmospheric drag influences both ejecta expansion processes and final ejecta distributions and to study relations of these effects to impact scale (bolide size) in general.

Methods

Theoretical Background

The numerical model is based on the SOVA multimaterial multidimensional hydrocode (Shuvalov 1999). In this study, we consider subaerial vertical impacts only, and apply a 2-D version of the code to projectiles of 100, 300, and 1000 m diameter. The 2-D approximation allows us to study the problem with high spatial resolution and high accuracy. Tables obtained with the ANEOS equation of state (Thompson and Lauson 1972) and input data for quartz from Melosh (2007) were used to describe thermodynamic properties of both target and projectile material. For simplicity, we used the same equation of state for both target and projectile material (quartz with density 2.67 g cm^{−3}). The air thermodynamics were described by the tabular equation of state (Kuznetsov 1965). Impact velocity was considered to be 18 km s^{−1}.

The code takes into account the influence of dry friction on the motion of disrupted rocks based on the approach developed by Melosh and Ivanov (1999). Acoustic fluidization as treated by Ivanov and Turtle (2001) is also incorporated. However, the target strength is not very important in evaluating the influence of impact scale on ejecta formation and distribution.

The transformation of solid and melted bulk ejecta into discrete particles is a principal point of the model (Shuvalov 2003). This transformation occurs when the ejecta reach an altitude equal to twice the projectile diameter. At this altitude, the ejecta have already reached their launch velocity and were not affected by the atmospheric drag. The portion of ejecta that is totally or partially evaporated during the impact is not transformed into discrete particles and is considered as a continuous medium. The partially evaporated material is believed to be the main source of spherules (Johnson and Melosh 2012). The further motion of discrete particles was described in the context of standard equations of multiphase gas dynamics (see e.g., Valentine and Wohletz 1989), taking into account their interaction with surrounding air and target/projectile material not transformed into discrete particles. The equations are valid when ejecta volume concentration is small (<<1), and we can neglect the collisions between ejecta particles. This condition is violated in the lower part of ejecta curtain. However, at the stage of high-velocity ejection (when ejected particles rise to altitudes exceeding crater diameter), the ejecta move ballistically and their interaction with air is negligible. At altitudes higher than projectile diameter, the ejecta volume concentration becomes small (<<1). The equations of multiphase gas dynamics are not valid for the slow close ejecta forming a crater rim. However, most of these ejecta do not rise above the altitude equal to twice the projectile diameter and are not transformed to discrete particles.

To solve these equations, a method of representative particles (markers) is used (Teterev 1999). Each marker describes the motion of a great number (10^{5} to 10^{10}) of real grains having comparable velocities, sizes, temperatures, and trajectories. A total of 500,000 representative particles were used to follow the motion of ejected target material. In other words, we replace a great number (10^{5} to 10^{10}) of real grains by a smaller number (500,000) of typical grains having the same total mass. A specially elaborated implicit algorithm provides correct solutions both for large boulders, which experience almost no atmospheric drag and follow ballistic trajectories, and for small particles, which move with gas flow velocity. The method also provides a correct terminal settling velocity due to gravity (which depends on particle size). This procedure is described in more detail in Shuvalov (1999, 2003).

The size-distribution of the ejected material is defined from the data available for lunar craters as described in Lindström et al. (2005). The largest (from the entire crater) ejected rock fragment size d_{max_max} is estimated as

dmax_max=(25±12)D0.69±0.03,(1)

provided that the crater rim diameter, D, is measured in kilometers, and d_{max_max} is measured in meters. The size of the largest ejecta fragment d_{max} in any particular ejecta volume is considered to be different for the volumes of rocks ejected at different velocities

dmax(V)=dmax_max(V/VR)(2)

where V is the launch velocity of a volume under consideration and V_{R} is an ejecta velocity near the transient crater rim (minimal ejecta velocity):

VR=[(4/15)gRt]1/2,(3)

where g is the gravitational acceleration.

Relation 2 is used to determine a value of d_{max} for each ejected volume. Then a standard presentation of the size-distribution in the form (Melosh 1989)

N(m)=Cfm−b,(4)

is used. Here, m is the mass of a fragment, N(m) is the cumulative number of fragments with mass equal to or greater than m, and b =0.85. The constant C_{f} is defined from the total mass of each ejected volume. We used 21 groups of particles with sizes from 1 μm to 10 m.

To study the influence of atmospheric air on the expansion of impact ejecta, we compared the results of detailed numerical simulations with the results of ballistic calculations and simple power approximations from Melosh (1989). At any moment, when solid ejecta are transformed into discrete particles, we know the initial velocities of ejection of each particle and, therefore, can calculate the final position of each particle on the assumption that it moves ballistically (without drag). These data were used to obtain the final ejecta distribution in ballistic approximation.

The influence of a wake created by the projectile during its flight through the atmosphere was taken into account. The air density within the wake is considerably less (by 1–2 orders of magnitude) than the density of the ambient air. Therefore, ejected material experiences much lower drag when it expands through the wake. The flight of the projectile from an altitude of 100 km (where air density becomes negligible) was modeled as described in Shuvalov and Trubetskaya (2007).

Results

Figure 1 shows the very initial stages of vertical impact of the 300 m diameter projectile: first at the actual instant of impact, then at 1 and 3 s after impact when a strong separation of ejecta by size is already evident. Larger particles experience weaker drag and move faster than smaller particles. The largest ejecta fragments move even faster than the shock wave in the atmosphere and penetrate outside the wake. Smaller particles (<1 cm) decelerate at the shock front where air density dramatically increases.

Ten seconds after the impact in the 300 m bolide simulation (Fig. 2), the cloud of large particles with individual sizes exceeding a few centimeters has expanded to altitudes and distances of some tens of kilometers. All the smaller particles, being decelerated near the shock front, follow the hydrodynamic flow. A portion of the vapor (potential spherules) accelerates upward through the wake and expands to very high altitudes (hundreds of kilometers). The remaining vapor condenses and gradually descends. The excavation phase has terminated at this point in the development.

In 1–2 min (Figs. 2 and 3), the largest particles reach their maximum flight altitude (100–150 km) and start to descend. Some large fragments ejected at the late stage of excavation, having relatively small ejection velocities, have already settled to the ground surface. Centimeter-sized ejecta may have expanded to a distance of about 40 km, but they too have begun to settle.

Five minutes after impact (Fig. 3), most meter- to centimeter-sized ejecta have been deposited (at distances up to 200 km). Millimeter-sized ejecta also begin to settle, while smaller particles reach their maximum altitude and continue to expand horizontally.

Figures 1-3 show that all ejecta can be roughly divided into two parts: “ballistic” ejecta and “convective” ejecta. The ballistic ejecta move due to their initial velocity (obtained during ejection) and their travel path is slightly modified (decelerated) by atmospheric drag. Smaller particles experience stronger drag and travel for shorter distances than larger fragments that had the same ejection velocity. According to Schultz and Gault (1979), the ballistic ejecta are larger than the critical particle size. The convective ejecta (consisting of small particles) decelerate just after ejection at a short distance from the launching point and their motion is controlled by atmospheric flows. Smaller particles stay longer in the atmosphere and travel greater distances.

In other words, the ballistic ejecta are the ejecta with which the air interacts only slightly, while the convective ejecta motion is entirely defined by the air flow. The degree of particle/air interaction can be defined by the time/length of particle travel before deceleration. The length L can be estimated as r(ρ_{p}/ρ_{a}), where r is a particle size, and ρ_{p} and ρ_{a} are particle (i.e., ejecta) and air densities. If L is lesser than a flow scale R (the size of highly disturbed region), then the particle/air interaction is strong and the ejecta will be “convective,” if L ∼ R the interaction is small and the ejecta will be “ballistic.”

In the case under consideration (impact of a 300 m diameter projectile), meter and decimeter-sized particles represent the ballistic ejecta (L = 0.3–3 km), and millimeter-sized and smaller particles (L <1 m) represent “convective” ejecta. There is no abrupt boundary between these two types of ejecta because particle size and L/R ratio change continuously. The critical particle size increases with increasing crater size (Schultz and Gault 1979).

Ten minutes after the impact, most millimeter-sized ejecta have been deposited. The finest ejecta (1–10 μm) are concentrated at altitudes below approximately 25 km. Only a very small amount of ejecta would be able to penetrate to higher altitudes through the wake. Vapor rises considerably higher, to 100–150 km. The total mass of ejected material (in the 300 m case) equals approximately 270M, where M is a projectile mass. This value includes only ejecta fragments, which rise to an altitude above 600 m (two projectile diameters). The total mass of excavated material is consequently somewhat larger. In the case under consideration (10 min), the mass of ejecta still suspended in the air equals approximately 10M. The mass of vapor is about 0.01M. We should recall that the term vapor refers to a portion of ejecta that was totally or partially evaporated during the impact.

Figure 4 compares three cases, impact of a 100, 300, and 1000 m-sized projectile, all 10 min after impact. Ejecta size-distributions for all three cases can be described by the same power law, but the size of maximum fragment increases with scale. There is no qualitative difference between the 100 m diameter projectile case and a 300 m diameter projectile impact (Table 1). There is only some quantitative difference. In both cases, fine ejecta decelerate, no s in the air at a small distance from launching point, then rise to the stratosphere by air flows induced by the impacts. In the kilometer-scale impact, the mass of ejecta is so large that it moves the atmospheric air itself on its way to high altitudes. Thus, the atmosphere cannot decelerate even the fine ejecta and they consequently expand to the rarefied upper atmosphere. In the upper atmosphere, even fine ejecta move more or less ballistically and therefore may travel to high altitudes. This is why ejecta distributions for large impacts strongly and qualitatively differ from those for small impacts (at least 10 min after the impact). After near-ballistic flight in the rarefied upper atmosphere, the fine ejecta decelerate in the lower dense atmosphere (at altitudes below 100 km) and their subsequent evolution is defined by atmospheric flows.

The relative total ejecta mass M_{t} (106M for D = 1000 m, 270M for D = 300 m, and 590M for D = 100 m) decreases when impact scale increases because gravity prevents crater growth for large impacts and the ratio of crater size to projectile diameter decreases when impact scale increases (Melosh 1989). The relative mass M_{v} of vapor ejected to high (tens of kilometers) altitudes equal to 0.43M for D = 1000 m, 0.01M for D = 300 m, and none for D = 100 m, increases when the projectile size increases, as in large impacts vapor penetrates to high altitudes where condensation is weakened because of reduced pressure. In small impacts, the vapor decelerates at low altitudes and begins to condense under higher pressure. In the smallest impact under consideration (induced by a 100 m diameter projectile), all the vapor material decelerates at low altitudes, condenses, and transforms into ordinary impact melt.

The relative suspended (at 10 min after the impact) mass M_{s} decreases when impact scale increases because in larger impacts, the bulk of the ejected mass is contained in larger fragments, which settle in a shorter time. The maximum size of ejected fragment decreases when impact scale decreases (Melosh 1989), and a relative portion of small and long-living (in particular, living more than 10 min) particles increases. Figure 5 shows thicknesses δ of ejecta blanket produced by impacts of 300 and 1000 m diameter projectiles. For comparison, the results of ballistic approximation and simple power approximations are shown in addition to the numerical simulations of the present study. At each moment, when solid ejecta is transformed into discrete particles, we know the launching velocities of the discrete particles and can calculate their final positions neglecting atmospheric drag (i.e., the ejecta were considered to fly ballistically). The final positions were used to calculate a thickness of ejecta blanket in ballistic approximation. In other words, the “ballistic approximation” means that we use the calculated values of ejection (launch) velocities and assume ballistic flight without atmospheric drag. The power approximation was taken from Melosh (1989):

δ=A(r/Rt)−α(5)

where r is a distance from the crater center, R_{t} is a radius of transient crater, A is constant. One can see that in the 300 m case, the power approximation well fits the results obtained in ballistic calculations. The presence of air considerably diminishes thickness δ of ejecta at distances above 15 km (that is, at distances exceeding 3 crater diameters). In the 1000 m case, all three curves correlate well at distances up to 400 km. However, the power α in the power law equals 3 for the small impact and 3.25 for the 1000 m impact. In larger impacts, the thickness of the ejecta layer attenuates faster.

Figure 6 shows ejecta distributions for particles of different sizes. Large particles, with size exceeding 3 m, experience very small atmospheric drag and move almost ballistically. The detailed numerical simulations (taking into account air/particle interaction—in this case minimal) and ballistic approximation give approximately the same thickness of ejecta layer produced by ejecta of this size for both impacts (300 and 1000 m diameter projectiles).

The influence of atmospheric drag increases when we consider smaller ejecta fragments. One can see that centimeter-sized particles are strongly affected by atmospheric drag in the impact produced by a 300 m diameter projectile. In the large impact case (1000 m impactor), this influence is less prominent. An increase in impact scale decreases the influence of atmospheric drag because in small impacts, the atmosphere decelerates ejecta, whereas in large impacts, the ejecta accelerates the atmosphere. For millimeter-sized particles, the deviation from ballistic approximation becomes more evident. The ejecta blankets for impact produced by a 100 m diameter projectile behave very similarly to the D = 300 m alternative and are consequently not presented.

Unfortunately, the results presented here do not allow us to say much about long-time expansion and final deposition of “convective” ejecta since the simulations were terminated at 10 min after impact. We cannot prolong the simulations for hours and days to obtain the final ejecta deposits. The distributions of suspended ejecta 10 min after the impact are quite different in small impacts (induced by stony projectiles less than approximately 500 m) compared with large impacts. In the first case (small impacts), the ejecta are concentrated at altitudes of some tens of kilometers within a cloud a few tens of kilometers in size. Further evolution of the ejecta cloud is defined by weather conditions and final ejecta deposits should look similar to dust deposits produced by strong explosions and volcano eruptions. This distribution should be very anisotropic, depending on wind direction.

In the case of large impacts, the “convective” ejecta are ejected to very high altitudes (hundreds of kilometers) and are dispersed in a cloud 1000 km in size. Thus, one can expect more or less isotropic (“convective”) ejecta deposits.

The model does not consider in detail the process of spherule formation. It can only be suggested that target and projectile material totally or partially evaporated during the impact is an important source of globally distributed spherules. If this assumption is true, small impacts (produced by projectiles 100 m in size and smaller) are not expected to produce spherules expanding all over the world.

Discussion and Conclusion

The numerical simulations presented above demonstrate that impact scale has a considerable influence on ejecta expansion and deposition. In small impacts, the “convective” ejecta decelerate just after ejection at a short distance from impact point. Their further motion is controlled by atmospheric flows induced by impact (first minutes) and defined by local weather conditions (for hours, days, etc.). In large impacts, even “convective” ejecta move almost ballistically at the initial stage (being ejected into the rarefied upper atmosphere). The boundary between small and large impacts approximately corresponds to a projectile diameter of about 500 m, but depends on projectile density and velocity. At the late stage (hours and longer), the fine “convective” ejecta move with atmospheric flows in all (even large) impacts. The impact scale only slightly influences an expansion of “ballistic” ejecta consisting of large (meter-sized and larger) fragments. In this study, we considered vertical impacts only and had axisymmetric ejecta clouds. In oblique impacts, the ejecta expansion will be asymmetric, especially for high-velocity ejecta. However, the obliquity does not change the main conclusions because the presented results can be applied to any portion of ejecta expanding in any particular direction (in oblique impact).

Impact velocity considerably influences a mass of evaporated and partially evaporated rocks. An increase in impact velocity enhances a process of evaporation, but the impact velocity only slightly influences the process of ejecta–air interaction. An increase in impact velocity results in increased ejection velocity and, consequently, in increased distance and time of ejecta deceleration. For the fine (convective) ejecta, the distance and time remain minor and the influence of impact velocity (and ejection velocity) can be neglected. The ballistic ejecta expand to some larger distances. In other words, the impact velocity can quantitatively influence the size and mass of ejecta cloud, but does not change the boundary between small and large impacts. Target properties also significantly affect the ejection mass, angle, and velocity, but do not influence the character of ejecta–air interaction.

This difference in ejecta distribution between large and small impacts probably cannot be observed in the field, as the data on the fine and distal ejecta distributions are very scarce and not detailed enough. Continuous ejecta blankets around craters (with a size of a few crater diameters) are produced by “ballistic” ejecta and, consequently, are not strongly affected by the impacts' scale. This effect is more important for the problem of atmosphere contamination by fine dust (induced by impact) because mechanisms of fine ejecta expansion are different in large and small impacts. In small impacts, an asymmetry of ejecta expansion is defined by weather conditions only; in large impacts, an asymmetry is strongly affected by impact asymmetry (impact angle and direction of flight).

Our numerical simulations show that ballistic approximation better describes ejecta deposits around large craters than ejecta deposits around small craters. This is explained by a difference in ejecta size-distributions. In large impacts, the size-distribution is shifted to large sizes, and larger fragments experience smaller atmospheric drag.

In ballistic approximation (e.g., in impacts on planets or satellites without an atmosphere), time of ejecta deposition does not depend on impact scale because this time is defined solely by ejecta velocities (depending on impact velocity, but not on the scale). Our numerical simulations show that this conclusion is approximately valid for “ballistic” ejecta in impacts on planets surrounded by atmospheres. However, the “convective” ejecta in large impacts rise to higher altitudes and live longer than in small impacts.

Our study shows in addition that ejecta layer thickness decreases versus relative distance faster for large impacts than for small ones. In order to understand this difference, let us consider ejecta launched at velocities larger than u from craters with radii R_{1} and R_{2} > R_{1}. If we ignore the influence of gravity and atmospheric drag, then in both cases, we will have the same relative ejecta mass M with launch velocities exceeding u_{0} (self-similarity). Ejecta particles launched at an angle of 45° and velocity u_{0} travel for a distance r = u^{2}/g from the launch point. Here, g is gravitational acceleration. One can see that the relative ejecta mass M is deposited at a distance exceeding r in both cases; r does not depend on crater radius. The relative distance r/R_{2} (which is used in approximation (5)) from the larger crater is smaller than relative distance r/R_{1} from the smaller crater. Therefore, if we consider the same relative distance r/R, we will have a larger mass at distances exceeding this value in small impacts than in larger impacts. This means that in small impacts, a larger portion of ejecta is deposited at high distances. This simple estimate explains the strong decrease in ejecta layer thickness with distance around large craters. We should bear in mind that ejecta blankets around small craters are formed by slow ejecta because fast ejecta expands to very high relative distances r/R = u^{2}/g/R. The slow ejecta are launched at a late stage of impact and are strongly affected by gravity. For this reason, the estimate considered above is not correct for small craters for which the ejecta blanket is geometrically similar (Housen et al. 1983). Our simulation shows that ejecta blankets around craters produced by 100 and 300 m diameter projectiles are described by the power law with the same exponent α = 3. In the D = 1000 m case, the exponent is bigger, α = 3.25. In the D = 10 km alternative, the exponent α is bigger and changes from 3.5 for close ejecta to about 4–5 for distal ejecta (Shuvalov 2011).

Our simulations show that the mass of partially or totally evaporated target/projectile material ejected to high altitudes decreases when impact scale decreases. This probably means that small impacts could not produce a considerable amount of spherules expanding to high distances (the vapor condenses and settles near or within the crater), a question that needs more detailed investigation.

Acknowledgments

Dieter Stöffler and Adrian Read kindly commented on an earlier draft of the manuscript. The detailed comments of editor Alex Deutsch, referee Kai Wünnemann, and one anonymous reviewer are highly appreciated.